I've highlighted 8, 14 and 22 units as those are the 50%, 75% and 90% marks. I have also included the desperate mulligan line to show how mulliganing to 6 decreases the chances of pledging (and likely many other things) by about 10% across the board.

This chart is nice for determining what percentages of hands of the 3 types have pledge available, but if I'm building a deck that truly cares about pledge, I'm going to mulligan any opening hand that doesn't have a pledge unit. If I'm going really deep, I might do a desperate mulligan looking for pledge. This means our computation looks like:

Conveniently, the 50%, 75% and 90% marks are at 4, 8, and 12 units, that is, 1, 2, or 3 playsets. Having 16 units leads to pledging 96% of the time, 20 units pledges in 98.5% of games, and 24 units will result in a pledge a near-certain 99.5% of the time. We can see that being willing to desperately mulligan will dramatically improve chances to find a pledge.

Sidebar: adding 4 pledge units lets your deck curve out as if it had about 2 more sigils in it. 8 units is about 4 sigils and so on. I don't want to go into the math here (there's a future article I have brewing), but for now, take it as given.

As a result, if you are building a pledge matters deck, you likely won't need more than 25 power in the deck (although I'd probably include some number of seek power or cargo depending on where your deck's curve and influence requirements are).

Now that we have a feel for how many pledge units to include, what about also drawing a pledge payoff card?

The following tables shows the probabilities of "getting paid off" for pledging after 5 and 10 draws. More precisely, it is the probability of having a pledge unit at the start of the game (I presume you play it as a sigil), and then either having a pledge payoff in your opening hand or drawing it after 5/10 turns. These also were computed with our friend, the hypergeometric distribution, combined with the results of the second chart above.

Assumptions:

These charts are for the "worst" case of being on the play, but if you are on the draw, the probabilities can be 0-2 percentage points higher.

You will mulligan your opening hand if it does not have pledge (but not desperately mulligan).

25 power deck.

What I find interesting here is the interplay between pledge and payoff numbers. Specifically, adding more payoffs into a deck will make pledge matters more consistent. For example, looking at 12 pledge units and 8 payoffs after 5 draws, one might wonder: should I add more pledge units or more payoffs? Notice that adding 2 more pledge cards only yields a 3 percentage point increase to the "paid off metric", but adding 2 more payoffs yields a 7 percentage point increase.

As another example, suppose you have 12 pledge units and 4 payoffs - should you put a payoff in the market? The numbers give a resounding yes! Putting 3 copies in the main deck with 1 in the market and 4 merchants is effectively 7 copies of the payoff, which yields a 19 percentage point increase.

The chart also shows that adding more pledge units past 20 has almost no effect and 16 units is probably fine, since going from 16 -> 20 only increases the "paid off" percentage by 2-3 points.

For all the mathy details, check out this Google Sheet with all the calculations and charts. Feel free to make a copy and modify some of the values to play around with things.

If you find a bug in my math or have some improvements in mind, feel free to let me know! You can comment on the Reddit thread, ping me on Discord at spaceonaut #4197, or find me in game at spaceonaut+0764. Happy brewing!